C₃×#1721 #72436

order	384
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₈

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

242 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 19 points:

(2,5)(3,7)(6,8)(10,13)(11,14)(15,16)(17,18,19);(1,2,3,5,4,6,7,8);(2,6)(5,8)(9,10,14,16,12,15,11,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5012);